On submanifolds of highly negatively curved spaces

We prove spectral, stochastic and mean curvature estimates for complete m-submanifolds phi : M -> N of n-manifolds with a pole N in terms of the comparison isoperimetric ratio I-m and the extrinsic radius r(phi) <= infinity. Our proof holds for the bounded case r(phi) <= infinity, recovering the known results, as well as for the unbounded case r(phi) = infinity. In both cases, the fundamental ingredient in these estimates is the integrability over ( 0, r(phi)) of the inverse I-m(-1) of the comparison isoperimetric radius. When r(phi) = infinity, this condition is guaranteed if N is highly negatively curved.